2.8 The F Distribution

The F distribution is a continuous probability distribution that is helpful in testing hypotheses about variances. The F distribution will be used in Chapter 4 when regression models are tested for significance. Figure 2.14 provides a graph of the F distribution. As with a graph for any continuous distribution, the area underneath the curve represents probability. Note that for a large value of F, the probability is very small.

A line begins at the graph’s origin and steeply ascends up and to the right and then begins to gently and smoothly descend back toward the x axis.

Figure 2.14 Full Alternative Text

The F statistic is the ratio of two sample variances from independent normal distributions. Every F distribution has two sets of degrees of freedom associated with it. One of the degrees of freedom is associated with the numerator of the ratio, and the other is associated with the denominator of the ratio. The degrees of freedom are based on the sample sizes used in calculating the numerator and denominator.

Appendix D provides values of F associated with the upper tail of the distribution for certain probabilities (denoted by $α$ $α$ ) and degrees of freedom for the numerator $({df}_{1})$ $({df}_{1})$ and degrees of freedom for the denominator $({df}_{2}) .$ $({df}_{2}) .$

To find the F value that is associated with a particular probability and degrees of freedom, refer to Appendix D. The following notation will be used:

\begin{array}{l} {df}_{1} & = & degrees of freedom for the numerator \\ {df}_{2} & = & degrees of freedom for the denominator \end{array}

$\begin{array}{l} {df}_{1} & = & degrees of freedom for the numerator \\ {df}_{2} & = & degrees of freedom for the denominator \end{array}$

Consider the following example:

{df}_{1} = 5

${df}_{1} = 5$

{df}_{2} = 6

${df}_{2} = 6$

α = 0.05

$α = 0.05$

A screenshot of the Excel calculations that show F distribution with d f one and d f two degrees of freedom. — Program 2.4A Excel 2016 Output for the F Distribution

Figure 2.4A Full Alternative Text

Screenshot showing the formulas used to find the F negative value and probability when capital F is greater than lowercase f. — Program 2.4B Functions in an Excel 2016 Spreadsheet for the F Distribution

Figure 2.4B Full Alternative Text

From Appendix D, we get

F_{α, df 1, df 2} = F_{0.05, 5, 6} = 4.39

$F_{α, df 1, df 2} = F_{0.05, 5, 6} = 4.39$

This means

P (F > 4.39) = 0.05

$P (F > 4.39) = 0.05$

The probability is very low (only 5%) that the F value will exceed 4.39. There is a 95% probability that it will not exceed 4.39. This is illustrated in Figure 2.15. Appendix D also provides F values associated with $α = 0.01.$ $α = 0.01.$ Programs 2.4A and 2.4B illustrate Excel functions for the F distribution.

The graph and line from the previous figure is repeated here. The shaded area is labelled zero point zero five and the segmenting line is labelled F equals four point three nine. — Figure 2.15 F Value for 0.05 Probability with 5 and 6 Degrees of Freedom

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Table of Contents for
2.8 The F Distribution

2.8 The F Distribution

Figure 2.14 The F Distribution

Program 2.4A Excel 2016 Output for the F Distribution

Program 2.4B Functions in an Excel 2016 Spreadsheet for the F Distribution

Figure 2.15 F Value for 0.05 Probability with 5 and 6 Degrees of Freedom

Table of Contents for 2.8 The F Distribution

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Table of Contents for
2.8 The F Distribution